A positive bound state for an asymptotically linear or superlinear Schrödinger equation

We establish existence of a positive solution to the stationary nonlinear Schrödinger equation−Δu+V(x)u=f(u),u∈H1(RN), in situations where this problem does not have a ground state. We consider general superlinear nonlinearities and asymptotically linear ones, as well.